Purdue University: ECE438  Digital Signal Processing with Applications
1
ECE438  Laboratory 7:
DiscreteTime Random Processes (Week 1)
October 6, 2010
1
Introduction
Many of the phenomena that occur in nature have uncertainty and are best characterized
statistically as random processes.
For example, the thermal noise in electronic circuits,
radar detection, and games of chance are best modeled and analyzed in terms of statistical
averages.
This lab will cover some basic methods of analyzing random processes. Section 2 reviews
some basic definitions and terminology associated with random variables, observations, and
estimation. Section 3 investigates a common estimate of the cumulative distribution function.
Section 4 discusses the problem of transforming a random variable so that it has a given
distribution, and lastly, Section 5 illustrates how the
histogram
may be used to estimate the
probability density function.
Note that this lab assumes an introductory background in probability theory.
Some
review is provided, but it is unfeasible to develop the theory in detail. A secondary reference
such as [1] is strongly encouraged.
Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman,
School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494
0340; [email protected]
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Purdue University: ECE438  Digital Signal Processing with Applications
2
2
Random Variables
The following section contains an abbreviated review of some of the basic definitions associ
ated with random variables. Then we will discuss the concept of an
observation
of a random
event, and introduce the notion of an
estimator
.
2.1
Basic Definitions
A
random variable
is a function that maps a set of possible outcomes of a random experiment
into a set of real numbers.
The probability of an event can then be interpreted as the
probability that the random variable will take on a value in a corresponding subset of the
real line. This allows a fully numerical approach to modeling probabilistic behavior.
A very important function used to characterize a random variable is the
cumulative
distribution function (CDF)
, defined as
F
X
(
x
) =
P
(
X
≤
x
)
x
∈
(
−∞
,
∞
)
.
(1)
Here, X is the random variable, and
F
X
(
x
) is the probability that X will take on a value in
the interval (
−∞
, x
]. It is important to realize that
x
is simply a dummy variable for the
function
F
X
(
x
), and is therefore not random at all.
The derivative of the cumulative distribution function, if it exists, is known as the
prob
ability density
function, denoted as
f
X
(
x
).
By the fundamental theorem of calculus, the
probability density has the following property:
integraldisplay
t
1
t
0
f
X
(
x
)
dx
=
F
X
(
t
1
)
−
F
X
(
t
0
)
(2)
=
P
(
t
0
< X
≤
t
1
)
.
Since the probability that
X
lies in the interval (
−∞
,
∞
) equals one, the entire area under
the density function must also equal one.
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 Spring '08
 Staff
 Digital Signal Processing, Signal Processing, Probability theory, Purdue University

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